Question 1123692
<br>
{{{drawing(400,400,-1,6,-1,6,
line(0,1,3,1),line(3,1,3,4),line(3,4,0,4),line(0,4,0,1),
line(1,2,4,2),line(4,2,4,5),line(4,5,1,5),line(1,5,1,2),
line(2,0,5,0),line(5,0,5,3),line(5,3,2,3),line(2,3,2,0),
locate(-.5,2.5,J),locate(2.5,5.5,K),locate(3.5,-.5,L),
locate(2.5,2.5,a),locate(2.5,1.5,b),locate(1.5,3.5,c),locate(3.5,2.5,d),
locate(0.5,2.5,e),locate(3.5,4.5,f),locate(4,1,g)
)}}}<br>
Refer to the above diagram for the discussion below.<br>
I will use whole numbers (percentages) to avoid having to write all the decimal numbers.<br>
The givens are:
(1) Pr(J) = 29  (a+b+c+e)
(2) Pr(K) = 23  (a+c+d+f)
(3) Pr(L) = 39  (a+b+d+g)
(4) Pr(JK) = 13  (a+c)
(5) Pr(J'L') = 51  (we can't really use this, because it includes regions outside of the union of J, K, and L)
(6) Pr(K'L) = 28  (b+g)<br>
It turns out we need to introduce two variables:
(7) a = x
(8) b = y<br>
Then we can do some calculations to put expressions involving x and y in regions c, d, e, f, and g.<br>
(9) g = 28-y  (from (6) and (8)
(10) d = 11-x  (from 3, 7, 8, and 9)
(11) c = 13-x  (from 4 and 7)
(12) e = 16-y  (from 1, 7, 8, and 11)
(13) f = x-1  (from 2, 7, 10, and 11)<br>
And now we can answer the questions that are asked.<br>
What is Pr(J&#8746;K)?  Answer: a+b+c+d+e+f = 39; so Pr(J&#8746;K) = 0.39.<br>
What is Pr(J&#8745;L)?  Answer: a+b = x+y.<br>
It is not possible, with the given information, to find a numerical value of Pr(J&#8745;L).  (10) and (13) above restrict the possible values of x to between 1 and 11; (12) restricts the possible values of y to between 0 and 16.  But any combination of those values of x and y yields a solution to the problem.<br>
What is Pr(K&#8745;L&#8242;)?  Answer: c+f = 12; so Pr(K&#8745;L&#8242;) = 0.12.